Modelling and parameter estimation of paper machine drying sections

Digi tal Computer Applications to Process Control, Van Nauta Lemke, ed. © IFAC and North-Holland Publishing Company (1977)

MODELLI GAD PARAMETER ESTIMATIO A. LEMAITRE *

A7-2

OF PAPER MACHI E DRYI lG SECTIO S

M. PERRO **

C. FOULARD *

*Laboratoire d'Automatique de 1'1 PG - Grenoble, France. **Centre Technique du Papier - Grenoble, France.

ABSTRACT Drying is a very important phase of papermaking. Improving its economic operation is then of great interest. Recent process analysis techniques by modelling, identification and simulation offer new perspectives for such studies.

Clotht~

This paper describes a mathematical model based on a theoretical analysis of the physical phenomena that occur in multicylinder paper machine drying sections. The main design and working parameters of these processes are taken into account and computer integration of the equations leads to an overall model of the sheet drying process which is easily workable. This model is the basis of a parameter estimation method which has been developed to evaluate the heat and mass transfer coefficient values in industrial equipment. Results from a pilot paper machine are discussed and the influence of some working parameters on the transfer coefficient values are shown.

blowing

Pocket

Felt drier FIGURE 1 : A multicylinder drying section Drying is a very important phase of papermaking since it has a strong influence on the final paper quality as well as on the economics of the whole process. It is the last part of the water removal phase, in which most of its mechanical properties are conferred to the paper sheet ; besides both the energy consumption and investment costs required by this process are high. Improving the economic running of paper machine drying sections is then of ver great interest and this subject has always been much studied. However the recent process analysis techniques, namely modelling, identification and simulation, have not yet been widely used although they offer new perspectives for such studies. The present contributions endea ours to act in that direction.

end, this moisture must be about 5 to 10 per cent. The whole drying section is set in a hood so that the atmosphere in which the evaporation takes place can be controlled by action on the insulated and extracted air flows. The section is fed with steam at a pressure of 3 to 10 bars. Since the temperature set-points of the cylinders rise from the beginning to the end of the section, the condensate of every drier is flashed and the steam thus produced is used by the former driers.

A multicylinder paper machine drying sections is made of steam heated cylinders on which the sheet is successivel applied by clothing (figure 1). The sheet moisture at the entrance of the drying sections is generally 60 to 70 per cent (water weight/total weight). At the

umerous studies have been carried out concerning the drying sections and the theoretical knowledge about the mechanisms that occur is quite large. any authors have analysed the heat and mass transfer conditions to which the sheet is submitted during its trip in the

357

ODELLI G

358

o PARA ETER ESTI 1ATIO

drying sections [I, 2, 3J. Very detailed descriptions of the migration of water and vapor inside the sheet have also been proposed [4, 5, 6J. However the design of the drying sections has so far been largely based on empirical rules and the optimum performances as well as the corresponding working conditions are not very well known. As a matter of fact few authors used these theoretical results to analyse the whole process behaviour [7J. any simplifications must be introduced to obtain workable models and achieve the complete computer simulations from which then the relative influence of ach design or running parameter can be seen. The elaboration of such a mathematical model of the sheet drying process is pres nt d here. The main simplifing hypotheses are as follows : two typical posltlons of the paperweb are distinguished: web on th cylinder and web in the draw - temperature and moisture is supposed to be uniform through the web. Only variations in the machine direction (x) are considered;

OF PAPER

This model has of course been used for computer simulations. But it is also the basis of a parameter estimation method which has been developed to evaluate the heat and mass transfer coefficients in industrial quipment. Results of this method will be described in the case of a pilot paper machine drying section.

A7-2

The sheet receives heat from the cylinder (of course only in the case of web on cylinder) e(8 c -

f)·

It exchanges heat with the clothing (web on the cylinder)or with the ambient air (web in the draw) ( --8 )· f

It also loses some heat b cause of evaporation of water. Calling B the evaporation rate, the loss is (r being the latent heat of vaporization of water) B. r (8 ) . f

For the two typical positions of the web we have . web on the cylinder : heat transfer coefficient with the clothing,

h

= 8h

temperature of the clothing

if the input of the model is the cylinder sursurface temperature then c cy 8c 8 cy if the input of th model is the steam temperature inside the cylinder then v c 8v ·

web in the draw c

=0

but since the heat-exchange with the ambient air takes place on the two sides of the sheet,

2.

At the abscissa x in the machine direction, we consider a sheet element of length dx, width L and oving at the speed C (figure 2).

2.

10reover

2.2.

'?f(x+dx) Xf(x+dx) x

x+dx

Sheet ele ent on the c linder

a

( a = heat transfer coefficient between sheet and air) V

a

air temperature.

ater balance for the sh et element

The water inside the paper is removed by surface evaporation due to the difference of partial pressure of water between the sheet surface and th ambient medium. In the air, the evaporation rate is given by the Stefan diffusion la B

FIG RE 2

ECTIO S

2. I. Thermal balance for the sheet element

- all the internal phenomena (vapor diffusion, capillary transport) are considered only with global relationships ; the drying section is in a steady state.

ACHI E DRYI G

=

RTf

P-p P log P- pf

ODELLI G

A7-2

o PARA ETER ESTI

p

vapor pressure

pf

vapor pressure at the sheet surfac

T

absolute temperature of the she t.

f

in the ambi nt medium.

Then the water balance for the sheet element gives :

P log ~ P-Pf

G

Web in th

(B)

draw : mass transfer co fficient in the draw ( vaporation from the two sid s of th sheet).

a

vapor pressure in th

p = Pa

pocke t air.

Web in the cylinder: there we assume that th Stephan law is always valid to describ the mass transfer betwe n web and clothing. The vapor pressure of the ambi nt medium p must then be taken as an equivalent vapor pressure set by th clothing in front of the sheet (Ph)' p

mass transfer between sheet and clothing.

TIO

OF PAPER ffiCHI E DRYI G SECTIO S

2.3. H groscopic porous media behaviour during drying. Water is fixed by an hygroscopic material with some nergy. This energy of absorption E induces a d crease of the vapor pressur in equilibrium with the material. The vapor pressure which is Ps when in equilibrium with water b comes Pf when in equilibrium with the hygroscopic material at th sam t mperatur T and we have : f RTf Ps E = - - log Pf The vapor being suppo molar mass 1.

h

a

d to be a perfect gas of

This nergy of absorption E appears only when th moisture X of th mat rial is low r than f a critical valu X and its value is th n co proportional to the inverse of th moisture X [8]. So f

E

x~o )

(~ X

K (T) o

f

Then the equilibrium sorption isotherms of the hygroscopic material can be describ d with the following relation :

k ( )( 1 __1_) f

o

Two kinds of clothing are considered : wire clothings which have a high permeability. We can assume that the ambient medium is al\a s the pocket air but the evaporation rate is reduced because the evaporation only takes plac on a given percentage of th sheet surface. This p rcentage of active sheet surface is higher as the porosity and permeabilit of the wire are higher. If we always refer to the real sheet surface then the effect of the wire is a reduction of the mass transfer parameter

359

Cl

)

Pf = Ps ( "'f e

\

Xf

X

co

If such a material is dried at constant temperature in air with a constant vapor pressure p , the vaporation rat aries during the dr'ing (figure 3).

-

B

'"'E

-........ ~

20

parameter characteristic of the \"ire

I J

-J

~

I

~

I

c

and,

0

10

r;) ~

c.

. felt clothings ~hich have a lo~ p rmeability but are water absorbent. In that cas the vapor pressure at the felt surface Ph will depend on the the felt.

e perature and

oisture of

These ~haracteristic of the water absorbent aterials (felt bein s'~ilar to paper) will be detailed separatel

~

X

1, .5

.oisture content X (kgH:O/kg dry solid) FIGCRE 3 : Dr.:ing of an h_ roscopic at rial. ain phase appear: - A surface evaporation phase. When the noisture of the material is high, the \apor pres ure Pf at the surface is qual to the aturated \apor pre sure at the a e temperature.Then Pf i constant (and also B) even when X decrease . f

T~o

360

ODELLI G

o

PARA ETER ESTI

~TIO

- An internal evaporation phase. When the moisture X becomes lower than the f critical moisture, Pf decreases and also B. The material surface can no longer be considered as a surface of free water and the vapor comes also from inside the material (internal evaporation) .

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OF PAPER MACHI E DRYI G SECTIO S

3. PARAMETERS ESTlMATIO In this model, the numerical values of many of the parameters which are required are unknown. This estimation of these parameters and of their variations in the drying section is necessary to carry on the study. In the first phase (cylinder), we need three different parameters : first, the coefficient of heat transfer from steam to web ,then the

The critical moisture, corresponding to the transition between the two drying phases, depends on the moisture gradient inside the material. This moisture gradient depends on the value of the evaporation rate (instantaneous and past). Then the critical moisture is higher when the evaporation rate is higher (figure 3). During drying the vapor pressure at the sheet surface is not only dependent on the temperature and moisture of the sheet, but also on the drying rate. However, this dynamic sorption isotherm can be described by the same relationship than for the equilibrium

(C)

v

coefficient of heat transf r from web to felt On and finally the coefficient of mass transfer

Bh

between sheet and dryer clothing.

It is possible to ignore the coefficient a h ; indeed the temperatures of the felt and the sheet are almost equal and the heat transfer is negligible. Computer simulations show that the value of this coefficient can vary over a wide range and has still a negligible effect on the drying process. In the draw, two parameters are required : the coefficients of heat and mass transfer a and B between the sheet a

a

and the air in the pocket. The transfer mechanisms being convective, it is possible, in this case, to assume that Colburn analogy is valid between the heat transfer coefficient a and a

the mass transfer coefficient B in the draw a

k

and X depend now also on the evaporation d cf rate and have different values than for the equilibrium sorption isotherms.

2.4. Complete sheet drying model For the sheet element that we have considered, the complete model is constituted by equations A and B with the transfer coefficients corresponding to the two web positions (cylinder and draw). The behaviour of the paper during drying is taken into account by relation (C). Finally the clothing nature is taken into account as described in 2.2., an other relation as (C) being used if the clothing is a felt. The integration of those differential equations on the cylinder and then in the draw is successively done for all the driers from the beginning to the end of the drying section. It must of course take into account the particular value of the para eters for each drier (as cylinder surface temperature or steam temperature, temperature and vapor pressure of the pocket air, nature and physical state of the clothing). This integration, quite easy to perform b means of a digital computer, leads to the complete sheet drying odel.

Fundamental knowledge concerning all these coefficients is not very large and the determination of their numerical values requires experimental work.

3.1. Experiments The experimental work has been carried out on the pilot paper machine of the "Centre Technique du Papier", Grenoble, France. This machine is 0,5 m wide and its highest speed is 120 m/mn. It is fitted with a process computer. The dryer section consists of 16 paper dryers (0,50 m diameter), The length of the draw is 0,50 m. All the dryers are clothed with felts. The pockets are ventilated by Madeleine hot-air blo ing rolls, and the air is extracted by an open hood. Measuring the sheet temperature during drying is not an easy task, 0\ ing to space restrictions in the pockets. A contact thermometer (S EMA) is used and two d if feren t easures a re taken on each dra\ . The posltlons of the two measuring points are not exactly at the beginning and at the end of the draw, but respectively at about one-fifth and four-fifths of its length. The sheet oisture content is determined by means ot paper samples taken on each draw at the end of the run.

A7-2

10DELLI G

D PARN1ETER ESTI tATIO

OF P PER

The samples ar roughly as long as the draw and therefor , the obtain d moisture content is equal to the average valu in the draw. The steam pressure of ach cylinder is measured after the steam f ding valv with 16 pressure sensors. Ev ry cylinder surface temperature as well as the clothing t mperature is measured with the same thermometer as used for the she t. Since th other characteristics of the felts (in relation (C), § 2.3.) cannot be measured, felts are assumed to be wires. Then the mass transfer coefficient h is a wire equivalent transfer coefficient. Its value can be high, if the felt is well dried, because the real value of Ph' that we cannot compute fr m the

CHI E DRYI G

361

ECTIO S

Drier i of th

pilot

machine.

(measured)

measurements, is then very low compared to the assumed value Pa' The wet and dry bulb temp rature of the air inside each dryer pocket are obtained from a psychrometer. We also measure the condensat flow of each dryer. Th experimental conditions of the two first r ference runs ar as follows Pulp..................

Kraft (20 0 SR)

Speed. . . . . . . . . . . . . . . . .

50 m/mn

Basis weight..........

92 g/m 2

Steam pressure........

2,5 kg/cm 2

In these runs clothing tension-stresses are as high as possible. On the contrary, in the third run, they are set at a low value to show the effect of clothing upon the transfer coefficients. The other operating conditions are kept constant, in order to compare the results of different runs.

1

8

~

(computed)

--, I I

8f i + 1

-+-~

: Xf i + 1

...

L

_

Simulated drier i

3.2. Parameters estimation methods From comparison between experimental data and simulations results, it is possible to estimate, cylinder by cylinder, the three coeffici nts tv' cy and for instance ta' by means of the-

FIGURE 4

Identification algorithm

following algorithm which is quite classical (figure ). This identification crit rion has a quadratic form 3 J

I j=1

\ i th _j calculated

a ured and a.

J

weighting coeffici nt.

m asured

.1any non linear programming methods have been t sted but some difficulties appeared concerning th algorithm convergenc . The isocriteria curves are very prolate and the otion of the algorithm is ver difficult on the resolution ridge, where the optimu of the function lies. hhen we use such ethods as for 'nstance 11 Gr ad i en t 11 0 r "Part a nI', \-of e nee d ate a c h s t e p the value of the criterion gradient. Because the equations of th system are coupled and non linear, it is not possible to get an analytic for of thi gradient; it is then evaluated with a first order approxi ation ~hich is very inaccurate near the resolution ridge. The ~otion of the current point of the algorithm is therefore very difficult and it stops before the optimum is reached. To overcome these

362

ODELLI G

D PARA ETER

STI ATIO

difficulties we have chos n more simple methods using "direct search" techniques without computing th gradient as HOOK and JEEVES method or as the rotating ax s m thods of ROSE BROCK. \ use the first one, which is not v ry fast but gives good r sults.

. RESULTS Figur 5 shows th variations of sheet t mperature, moisture cont nt and evaporation rate along the drying section, obtained by a computer simulation of the refer nce run. We can se that most of evaporation occurs on the cylinder. This is due, in this case, to the very low sp ed of th paper machin . The three transfer coefficients estimated for th se reference runs are plotted in figure 6. The value found for th heat transf r coefficient is 400 kcal/h.m 2 oC at the beginning of the d~yer section. Then it reaches 800 kcal/h.m 2 oC after the pr heating of the sheet and falls regularly to 200 kcal/h.m 2 oC on the last dryers. These values agree with several research workers results [9J [10J. The decrease of this co fficient at the end of drying is due to the decrease of the heat transfer coeffici nt with the moisture content of the web. The temp~rature of the outside surface of the dryer was also measured : hence the coefficient cy can be estimated with the same identification methods. These results are shown on figure 7, where w can see that cy is really responsible for the decreas of c Indeed, the resistance to heat transmission between web and cylinder surface decreases when the moisture content of the sheet is high and the contact more intimate. The numerical values of icy vary from 300 to 2 300 kacl/h.m 2 oC. The two other curv s plotted in figures 6 and 7 show the variations of the mass transfer coefficients hand a' The results are quite the same for these two different identifications (6 and 7) : h' the mass transfer coefficient between web and clothing, reaches a maximum value of 150 m/h during the preheating period and then decreases steadily along the drying section. At the best of our knowledge so far there has not peen an publication concerning these mass transfer coefficient values and then no comparison is yet possible. The ph sical interpretations of these values is quite difficult because their esti ation has been made \ ithout taking into account the exact condition of the felt and then the are global values of the felt efficacit y (or values of· its \ ire equivalent). Figures 6 and 7 show that, the mass transfer coefficient between air and web is nearl constant along t~e drying section. This is not surprising since a depends mainly on the entilation conditions (air speed at the sheet surface) \hich are constant along the machine because fix d by the machine speed. The a erage value of a in these runs is 20 /h and the corresponding alue of

~CHI

OF P PER

A7-2

E DRYI G 0ECTIO S

a' obtained by m ans of Colburn analogy is about 5 kcal/h.m 2 oC. Figure 8 shows the comparison of heat flows on the cylinders, first computed from sheet temperature and moisture content measurements and second obtained by means of condensate flow measurements. The difference between th two curves is due to the loss of heat in the cylinder and to the condensing of Dlowthrough steam .

Q-A

/

L.\

./

/~

~

\~

l,/ \ u'

V

.c

A~.-A~

\

/'

0

~

0/ \

e-

:3

0

"0 "0 )-I

;::j Cl)

.w Cil

::r:

;::j

c..

E ()

~

0

FIGURE 8

\ 0

'\

.w

E

6

\

bO

~

\ 0-0,

~

M

~

\

N

~

~ ,/\

5

0

10

Heat flow on each cylinder (computed and measured)

The accuracy of these results depends greatly on the measurements accuracy and reproducibility (particularly the exact measuring point position of the sheet temperature). Assuming the relative error in sheet temperature and moisture content to be 1 %, computer simulations show variations in the three transfer coefficients , and of respectivel 15 %, v 15 % and 20 Thisaunaccuracy is mainly responsible for th serrated shape of the curves (figures 6, 7, 9). The influence of clothing tension-stress is shown in figure 9. The main effect of the decrease of clothing tension is a decreased heat transfer coefficient . The resistance to the heat transmission frg~ the cylinder surface to the w b incr ases, because the contact betwe n the two surfaces is not good and the thickness of the air film is greater. This result shm s what are the possibilities of this para eter esti ation method.

A mathematical model of a co plete dr ing section has been developped from a theoretical anal sis of this process. Some simplifications in the description of the phenomena are introduced which lead to an overall model easil workable ; but the main elements of the phenomena are preserved so that this odel ma be

A7-2

ODELLI GAD PA

ETER ESTI1ATIO

us ful and particularly so that it can show the relativ effect of some design and running param t rs. This model is the basis of a parameter estimation m thod to evaluate the heat and mass transfer coeffici nt of industrial processes. The xperimentations on a pilot paper machine show that the main limitation conc rning the interpretation of the results is due to the measurem nt accuracy. One of the future d v lopm nts of this study will b the improvem nt of the estimation accuracy, first with mor accurate measurem nts but also with m asuring methods which will b mor reprodu ibl and asier to p rform (particularly for th sh t moistur ). Th n the stimation of r al heat and mass transf r coefficients of industrial equipm nts will b possible and v ry useful to analyse their performances and compare them taking into account the working conditions and not only an ov rall balance of th thermal and mass flow-rates. Another objectiv of this study is to search, by means of computer simulations, th optimum valu s of the design and running parameters of such processes. Th n this model could b come a useful tool for the paper machin us rs and design rs.

OF PAPER

[8J

[]J A.H.

sp cifi

estimation criterion.

x

abscissa in the machine direction.

U

speed of the w b.

L

width of the web. t mp rature of the web (absolute temperatur : T ). f moisture of th w b (water weight/fiber weight).

critical moisture of the web.

Xc h

cri t ic a 1 mo i s tu reo f the f e 1 t .

cy cy v

(5)

[4J W.D. BAI ES 0

f Canada 7 a

(3)

[7J R.L.C. K.. TIGHT and

L.A. KIRK

International \at r r moval S posium British Paper and Board Industr Federation. . larch 1975 - Lond n.

surface temp ratur

n w band

of cylinder.

heat transfer co fficient betwe n steam and w b. t mp rature of the st am insid cylind r.

the

heat transfer co ffici nt b tw and air.

n sh et

ambi nt air te p rature.

total surroundin

P

Pu 1 p and Pap r .1aga z i ne of Canada 65 (12) T 537-T 549 (1964).

h at transf r co fficient b tw cylind r.

vapour partial pr surface.

15 -]60 (]97/).

[6 J S. T. HA:

temperature of the clothing. heat transfer co fficient betw en \eb and clothing.

vapour partial pr ssure in the ambient air.

[5J F.T. HARTLEY and R.J. RICHARDS TAPPI 57:

basis weight.

x: c .+=. .

[3J J.A. DEPOY

Pulp and Pap r .1agazi ne (2) 5 -6/ (] 973) .

fibers

heat of water.

J

(2) 63-72 (]970).

Pulp and Pap r Magazine of Canada 73 67-7 (1972).

15-13 (1960).

S

sp cific h at

i

Pap ri Ja Puu 52

t de

and B... ORDGRE

Svensk Papp rstidning 63 : (2)

h

[2J O.J. LEHTIKOSKI

materiaux

Sand A.L. JA . SSO

[IOJ A.L. JA SSO

h

(196]).

ILLIERE

Svensk Papperstidning 60 : (]7) 621-631 (1957) .

1

: (8) 529-53

~1.A.

LEROY, . 1.C. PL CHET,

[9J O. BRA

ISSAl and D. HA . SE

TAPPI

363

E DRYI G SECTIO

Journees Internationales des gaz humid s. Institut Fran~ais des Combustibles l'Energie. Paris 25-27 juin 1959.

G REFERE CES

~1.R.

~CHI

a

sure at th

sh

pre sure.

mass transfer co fficient in the dra . vapour partial pre

ure at the felt surface.

mass transfer co fficient b t\een web and clothing. saturat d yap ur pressure at th temperature . coefficient of COLB R

analog .

considered

D PARA ETER ESTI ffiTIO

vDELLI G

364

OF PAPER

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A7-2

CHI E DRYI G SECTIO S

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FIGURE 5

Moisture ontent, teplp rature and evaporation rate of the sheet in the dry i n g sec t ion (c P1 put e r s i nJ U 1a t ion) .

800

..c l:i.

c::

......

l:i.

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90

E

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500

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FIG RE 6

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0

ffici

nts

v' 3

cy

'

r nUf!1ber

a in the dryin

secti)11

:J::>I -..J

I

o Mass transfer coefficient B

on the cylinder (m/h)

cy

• Mass transfer coefficient 'T1

W

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